Obstáculos superados pelos matemáticos no passado e vivenciados pelos alunos na atualidade : a polêmica multiplicação de números inteiros

In Mathematics literature some records highlight the difficulties encountered in the teaching-learning process of integers. In the past, and for a long time, many mathematicians have experienced and overcome such difficulties, which become epistemological obstacles imposed on the students and teachers nowadays. The present work comprises the results of a research conducted in the city of Natal, Brazil, in the first half of 2010, at a state school and at a federal university. It involved a total of 45 students: 20 middle high, 9 high school and 16 university students. The central aim of this study was to identify, on the one hand, which approach used for the justification of the multiplication between integers is better understood by the students and, on the other hand, the elements present in the justifications which contribute to surmount the epistemological obstacles in the processes of teaching and learning of integers. To that end, we tried to detect to which extent the epistemological obstacles faced by the students in the learning of integers get closer to the difficulties experienced by mathematicians throughout human history. Given the nature of our object of study, we have based the theoretical foundation of our research on works related to the daily life of Mathematics teaching, as well as on theorists who analyze the process of knowledge building. We conceived two research tools with the purpose of apprehending the following information about our subjects: school life; the diagnosis on the knowledge of integers and their operations, particularly the multiplication of two negative integers; the understanding of four different justifications, as elaborated by mathematicians, for the rule of signs in multiplication. Regarding the types of approach used to explain the rule of signs arithmetic, geometric, algebraic and axiomatic , we have identified in the fieldwork that, when multiplying two negative numbers, the students could better understand the arithmetic approach. Our findings indicate that the approach of the rule of signs which is considered by the majority of students to be the easiest one can be used to help understand the notion of unification of the number line, an obstacle widely known nowadays in the process of teaching-learning

Refletindo dificuldades de aprendizagem de alunos do ensino médio no estudo do equilíbrio químico

This research intends to identify the main conceptual mistakes committed by High School students when studying Chemical Equilibrium and also how the didactic books can influence in their formation. For the data - collecting we applied 150 questionnaires with 11 open questions e 4 restricted ones. Students from three different private schools in Natal-RN were used in this survey. The questionary about the book analysis tried to understand how the subject has been boarded, making it easier or not for the knowledge building. The books were chosen by being usually the most used ones at the chemistry classes in the schools in town. On the didactic book analysis we evaluated 12 items that tried to identify the clarity with which the books would bring up the subjects, the contextualization of the exercises, the analogies used, the amount and the quality of exercises, the connections between the subjects, and the retaken of important concepts. The questionary given to the students tried to get their knowledge about the concepts related to Chemical Equilibrium. The main mistakes found were: the definition of the state of equilibrium in a chemical reaction, the constant, the double headed arrow, confusions between mass and concentration, the representation of a microscopic reaction in equilibrium, partial pressure and total pressure, and the function of the catalyser. The results demonstrated a lot of insecurity or lack of knowledge about the subject when it was realized that there was a big number of blank questions and a considerable percentage of questions with conceptual errors. We concluded that those errors could be eased or even avoided if the didactic books and the teachers boarded the contents in a more conceptual form with the application of a largest number of good quality exercices, watching the building of definitions by the students, avoiding this way the formation of wrong concepts

“Currículo encarnado”: cartografia simbólica e afinidades pós-coloniais

This thesis has as main objective to reflect about the defining conceptual elements of the embodied curriculum concept, ident ified curriculum conception from the analysis about the rationality employed in the doctoral work of teacher Antonio Fernando Gouvê a da Silva, entitled The construction of the curriculum in popular critical perspective: the significant words to the context ualized practices , written and defended in the Postgraduate Program in Education: Curriculum, in the Pontifícia Catholic University of São Paulo, in 2004. We looked, also, on the problematization of the affinities between postcolonial theories – analytical perspectives towards the discussion about coloniality and their effect on contemporary social weaving – and the embodied curriculum concept. We argue that the reflections present in the aforementioned work bring an articulated curriculum concept based on three conceptual symbiotic elements, namely: negativity, dialogue and praxis, which, by endorsing the possibility of a curricular ingrained practice in the life context of the individuals, have some post - colonial inclinations that lead to the problematizat ion about the neo - colonial manifestations in curricular sphere, outlining a critique of the modus operandi of the colonialism, particularly, in its cultural and epistemic dimension from which the education is inseparable. For that, we used as methodologica l procedure the symbolic cartography, knowledge building strategy systematized by Boaventura de Sousa Santos, which allowed us to construct interpretive maps that enabled the symbolization of the universe which we longed to understand, that is, the concept ual categories mentioned above, which, in our view, underlie the concept of embodied curriculum. In this direction, we anchored ourselves in a meaningful dialogue with the theoretical approach of Paulo Freire and some of his interpreters regarding the disc ussion on curriculum, especially the reflections dev eloped by Antonio Fernando Gouvê a da Silva, and authors whose theoretical developments resonate in prospects for humanization, social justice and empowerment, among which we highlight: Theodor Adorno, Hug o Zemelman, Wilfred Carr, Adolfo Sánchez Vázquez, among others. We seek, in the same manner, on the contributions of authors considered post - colonial, as Hugo Achugar, Gayatri Spivak, Boaventura de Sousa Santos and Enrique Dussel, to name a few, the reason s why we consider the embodied curriculum as a place of political - pedagogical enunciation, conducive to an educational praxis that engages in a curricular work of reality translation in order to see what overwhelms it to, then, elicit the construction of a school curriculum as an awareness project for releasing in relation to what is unjust and inhumane. We consider, finally, that the success of this curriculum embodied translation implies a larger number of speakers mobilized in the production of knowledge that yearns for social emancipation and contribute to the enrichment of human capabilities as the maintenance of the life and the dignity of people.

Mobilização de competências na atividade informal do vendedor ambulante em praia de Natal (RN)

The central issue of this dissertation is to investigate the labor activity of beach hawker, in order to identify the main professional competencies mobilized in this activity, traversed by both the precariousness of the means of labor exercise, as for complex and structured routines. In the town of Natal (RN) the beaches serve as workplace for thousands of informal workers, who use various professional skills, translated into the ability to mobilize and articulate knowledge, skills and behaviors to solve problems in concrete work situations. This research therefore had as main objective to investigate the work of beach hawkers, trying to identify the core competencies mobilized for facing demands and obstacles in such a context. The beach of Ponta Negra (Natal-RN) was chosen as field of observation, in which a group of hawkers took part as voluntary subjects. Methodologically, quantitative and qualitative methods of production and analysis of data were combined in three stages. In the quantitative phase an occupational questionnaire was applied to a sample of 60 subjects, generating a set of data analyzed with quantitative univariate and multidimensional descriptive statistical tools, complemented by inferential statistical analysis. The results of this phase indicate a predominance of men sellers with salary varying in a range from one to two minimum wage Brazilian salary, age and education quite heterogeneous, extended working hours and the choice of only this activity and this beach throughout the year. Concurrently with this step of analysis, unsystematic observations of the activity of vendors were held and then driven to the technique of Instruction Impersonator with four chosen subjects. This phase had a clinicalinterpretive analysis, rooted in historical-cultural Vygotskian psychological perspective and in the french approach of skills and abilities. The main results point to several strategies for overcoming obstacles, use of technics anchored in everyday work experience and practical knowledge, building rules of conduct and collective mobilization of diverse professional skills similar to those found in formal work, such as business and time management, use of communicative tools, flexibility in problem solving, creativity and teamwork competence. We conclude that informality investigated in context can not be seen exclusively as a synonym of precariousness. It also covers skills and knowledge in a complex culture that situates informal labor in a complementary way with respect to formal work. This conclusion, therefore, contributes to overcome the notion of antinomy between formal and informal labor activity, since they both can be considered as a way to achieve job satisfaction, and even a personal representation of well done job, which is an important psychological generator of identity and social place.